12022
J. Phys. Chem. B 2002, 106, 12022-12030
Monte Carlo Simulations of Ag2+ and Ag2 in Aqueous Solution Vincent Dubois and Pierre Archirel* Laboratoire de Chimie-Physique UMR 8000, Baˆ t. 349, UniVersite´ Paris-Sud, 91405 Orsay Ce´ dex, France ReceiVed: January 11, 2002; In Final Form: September 3, 2002
We perform Monte Carlo simulations of Ag2+ and Ag2 in aqueous solution. We find that the hydration enthalpies are -5.56 eV for Ag2+ and -0.08 eV for Ag2, and that the redox potential is -2.0 V/NHE for the Ag2+/Ag2 couple. This redox potential is less negative than that of the Ag+/Ag couple, in agreement with the general trend, known for larger clusters. Our simulations qualitatively reproduce the absorption spectrum of Ag2+, observed in pulse radiolysis. We discuss the methods of calculation of the polarization and of the ionization free energy.
1. Introduction
2. VB Treatment of Ag2+
The process of metal agregation, initiated through irradiation of a solution of silver cations, has been investigated with the help of pulse radiolysis: absorption spectra of transient clusters have been recorded and redox potentials of Agn+/Agn couples have been evaluated.1,2 Interpretation of these measurements is a challenge for molecular simulation: calculation of absorption spectra requires the structure of clusters in solution and the position of excited states, calculation of the redox potentials requires ionization free energies. All these quantities are difficult to calculate. In a former work,3 we have obtained Monte Carlo values of the hydration enthalpies of Ag+ (-5.65 eV) and Ag (+0.05 eV), and of the redox potential of the Ag+Ag couple (-2.3 V/NHE). These values are in good agreement with the measurements of the hydration enthalpy of Ag+ (-5.50 eV) and redox potential: -1.9 V/NHE (electrochemical value) and -2.1 V/NHE (thermochemical value). In this work we used the Kozack and Jordan (KJ) water potential.8 In the present work we address the Monte Carlo study of the silver clusters Ag2+ and Ag2 in aqueous solution. We calculate the hydration enthalpy of both species and the redox potential of the Ag2+/Ag2 couple. Since the simulations require large amounts of computer time, we also work on accelerating the convergence of two time-consuming steps: the iterative calculation of the polarization energy and the calculation of the ionization free energy. In the sections 2 and 3 of this work we present the potentials for the Ag2+-solvent and Ag2-solvent interactions. In section 4 we investigate the convergence of the classical method for polarization. In section 5 we discuss the hydration enthalpies of Ag2+ and Ag2, the corresponding radial distribution functions (rdf) and the absorption spectrum of Ag2+. In section 6 we discuss the methods of calculation of the ionization free energy, and the value of the redox potential of the Ag2+/Ag2 couple. The ab initio calculations of Ag2 and Ag2+ and of their interaction with H2O are detailed in Appendix.
2.1. Introduction: Ag2+ in the Gas Phase and in Solution. In this treatment the wave function of Ag2+ is developed in the basis of two valence bond (VB) structures, with the electron localized on one of the atoms only:
* Corresponding author. E-mail:
[email protected]. Fax: 33 1 69 15 61 88.
Ψg,u(Ag2+) ) Ψ(AgAg+) ( Ψ(Ag+Ag)
(1)
where we have droped the normalization factors for convenience. Now considering that Ag2+ has one active electron only, we can use the equivalent orbital equations:
φg,u ) sl ( sr
(2)
where sl and sr stand for the 5s orbitals of the silver atoms. We note Agl, Agr, Ag+l, and Ag+r the left and right neutral and cation, Zl and Zr their nuclear charges ()1), ∆ the one electron ps + laplacian, Vps l and Vr the core pseudopotentials of Ag , R the distance of the two nuclei, and rl and rr the distances of the electron to these nuclei. With these notations the Hamiltonian of the bare one electron Ag2+ is given by
Zl Zr 1 1 ps - + hbare ) - ∆ + V ps l + Vr 2 rl rr R
(3)
In the basis of the orbitals sl and sr this Hamiltonian takes the form of a 2 × 2 matrix:
Hbare )
[ ]
a c , a ) 〈sl,hbaresl〉, c a
c ) 〈sl,hbaresr〉
(4)
We have found that using the genuine, i.e., non orthogonal, VB treatment of the bare cluster leads to severe difficulties in the subsequent treatment of solvation. Since Monte Carlo simulations require simple potentials, we therefore assume that the orbitals sl and sr are orthogonal. Within this assumption the two parameters of the intracluster potential, a and c, are given by
a ) (Eg + Eu)/2
c ) (Eg - Eu)/2
(5)
The a and c parameters therefore can be deduced for each value of R from the ab initio energies Eg and Eu of the two first states
10.1021/jp0200775 CCC: $22.00 © 2002 American Chemical Society Published on Web 10/24/2002
Ag2+ and Ag2 in Aqueous Solution
J. Phys. Chem. B, Vol. 106, No. 46, 2002 12023
of Ag2+. We have used the MP2 method with the core pseudopotential and basis set given in the Appendix, and a leastsquares fit. We now address the solvation of Ag2+: since we have Ag+water and Ag-water potentials available,3 the modifications the “dressing”sof the VB matrix of Ag2+ is a very natural way of taking solvation into account. This is easily seen on eq 4, where the diagonal terms of Hbare describe a localized electron, either in sl or in sr. We note E(Ag+w) and E(Agw) the pairwise Ag+-water and Ag-water potentials, and E3b(Ag+wiwj) the three-body Ag+-water potential of ref 3. It is convenient to make the dressing of the VB matrix in two steps:
Hbare )
[ ]
[
]
a + δal c + δc a c f Hdres ) f c a c + δc a + δar a + ∆al c + ∆c Hdres ) c + ∆c a + ∆ar
[
]
(6)
where the first dressing introduces one water molecule only, and the second introduces the whole environment. Keeping the dimension two of the matrix while introducing the solvent reflects the postulate that there is no charge transfer between the solute and the solvent. This is justified by the values of the ionization potentials of Ag (7.58 eV6) and H2O (12.6 eV6), which are largely different (by 5. eV). 2.2. Dressing of the diagonal of the VB Matrix. For the sake of simplicity, but with no loss of generality, we model water with a neutral ligand L, with nuclear charge ZL ()2) and two electrons in orbital l. Joining a ligand L to Ag2+ yields the three-electron cluster-water Hamiltonian:
hdres )
∑i
[
]
Z l Zr Z L 1 ps - ∆i + V ps (i) + V (i) - + l r 2 ril rir riL 1
∑ i
